Aircraft: B737
Phase of flight: Descent
Anomaly: Aircraft Equipment Problem Critical; Inflight Event / Encounter Bird / Animal

[We were] descending on the Arrival into ZZZ and on downwind for a visual approach to Runway XXR. As the aircraft began to descend from 9;000 feet to 8;000 feet at 210 knots; the crew heard a very loud bang. Both pilots suspected a bird strike. A few seconds later the ENG Master Caution came on then went back off. Both pilots agreed more than likely engine damage occurred; but a glance at the engine instruments showed both engines were normal operation. However; the autothrottles did kick off further indicating engine damage. The First Officer [advised] with Approach Control and requested priority handling for immediate landing. Approach requested to know the nature of the [event] time permitting. The FO (First Officer) informed Approach of suspected bird strike and engine surge suspected on number one engine.The Captain started the APU in anticipation of losing an engine. The FO suggested completing the Engine Surge Checklist. The Captain agreed and then also asked for the Single Engine Landing Checklist; which was performed. During turn to final the Captain noted an Airspeed Disagree Caution light. The FO confirmed the GPS ground speed was 178 knots; which matched the Captain's airspeed indicator. The FO's airspeed indicator was 30 knots low; thus; both pilots agreed the Captain's airspeed indicator was the reliable indicator. Since we were turning final and had visual contact with the runway; it was decided to continue the approach and to rely on Captain's airspeed indicator. We completed the Single Engine Before Landing Checklist prior to the final approach fix; and the Captain affected an approach to landing; flaps 30; at a faster than normal speed in anticipation of a possible engine failure. We touched down with no further incident.In regards to communication with the flight attendants and passengers; we asked the flight attendants to stay seated and that we would get back to them. The flight attendants did call us back later to inform us of visual damage due to bird strike. We thanked them and continued to focus on the approach to landing; we did not make an announcement to passengers. After landing it was noted that both engines were operating in the Alternate mode. Multiple bird strikes causing damage to both engines; leading edge slats and radome. We were abeam the airport on downwind; descending to 8;000 feet at 210 knots; and flaps one. We heard a very loud noise in the flight deck that sounded almost like an explosion. The aircraft also shuddered. My initial thought was we just blew an engine. I crossed checked the instruments and initially had an A/T LIM indication. The other engine indications looked fine. I then received a Master Caution and an ENG annunciator light...both engines had gone to ALT mode. I then thought we either took a bird or had a huge compressor stall. I've had compressor stalls before and; although loud and sometimes violent; not this violent. I had my FO (First Officer) [advise ATC] with immediate vectors to landing.I then received an IAS disagree light. I then; still not quite sure of our situation with multiple differing indications; directed the FO to get out the QRC for engine limit; surge; stall to ensure we had these areas covered in the event it was a stall; and ensure engine operation sufficient for landing. I had already disconnected the autothrottle. We went through the QRC; then the checklist; ensuring the engine was operating normally. With multiple abnormalities I was unsure of the exact nature of our damage or problem; but I had by now surmised that we had a major bird strike. This was also confirmed when the Flight Attendant called and said a Passenger saw a bird hit and lodge in the aircraft. Unsure of the extent of the damage or the possibility of increasing problems and/or structural damage; I elected to land immediately. I also had the FO run the One Engine Inoperative Landing Checklist as a precaution in case one of our engines failed; although I told him we'd remain at 30 flaps (I didn't want to introduce another change to flaps 15 at this point when not necessary). We started the APU and went to Flaps Inhibit and quickly reviewed our plan if the engine failed. I also had to quickly assess the IAS disagree. The maximum and minimum airspeed bars seemed much closer together than normal. I crosschecked my airspeed with my FO's and I was about 25 knots faster. I then crosschecked my airspeed with the aircraft groundspeed. My airspeed was validated by the groundspeed and the FO's was much lower. I used my airspeed and increased my VREF slightly to ensure I had enough speed and didn't even come close to the minimum speed bars...that I didn't fully trust at this point. We finished configuring and landed uneventfully. I did not get out a call to the passengers since we were very time compressed and I was focused on analyzing the problems and getting the aircraft on the ground.All ATC agencies were very helpful and expedited our recovery. The crew did an outstanding job. We both; the FO and I; had our hands full with multiple aircraft problems and the unknown of the full extent of our problems. I will tell you that having systems knowledge and some of the scenarios we have seen in the LOE (Line Operating Experience) (IAS Disagree for one) were huge assets for me to fall back on in this time compressed scenario. This is not something we normally train for. This scenario didn't fit the mold of a typical EP; so being able to quickly fall back on training and knowledge was instrumental in our quick recovery and landing. B737-700 flight crew reported several aircraft system malfunctions due to multiple bird strike on approach.